The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method Alexis Vasseur, and Yi Wang Department of Mathematics, University of Texas at Austin, Austin, TX 787, USA Institute of Applied Mathematics, AMSS, CAS, Beijing 9, China Abstract: We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier-Stokes-Fourier system. The method is based on the relative entropy method, and do not assume any smallness conditions on the discontinuity, nor on the BV norm of the initial data. It is proved that for any viscosity ν, the solution of the compressible Navier-Stokes-Fourier system with well prepared initial value converges, when the heat conductivity κ tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate κ. It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solutions for a system. Introduction We study the zero dissipation limit of the solution to the Navier-Stokes-Fourier system of a compressible heat-conducting gas in the Lagrangian coordinates in one-dimensional space R: t v x =, v t + p, ϑ x = ν v x x, e, ϑ + v t + p, ϑv x = κ ϑ x x + ν vv x x where the functions x, t >, vx, t, ϑx, t > represent the specific volume, velocity and the absolute temperature of the gas, respectively, p = p, ϑ, e = e, ϑ is the pressure and the internal energy satisfying the second law of thermodynamics, ν > is the viscosity constant and κ > is the coefficient of heat conduction. The work of A. F. Vasseur was partially supported by the NSF Grant DMS 94. E-mail: vasseur@math.utexas.edu. Y. Wang is supported by NSFC grant No. 736 and No. 36. E-mail: wangyi@amss.ac.cn..
If κ = ν = in., then the corresponding Euler system reads as t v x =, v t + p, ϑ x =, e, ϑ + v + pv x =. t. For the Euler equations., it is well-known that there are three basic wave patterns, i.e., shock and rarefaction waves in the genuinely nonlinear fields and contact discontinuities in the linearly degenerate field. If we consider the Riemann problem of compressible Euler system. with the following Riemann initial data, v, ϑx, = { L, v L, ϑ L x <, R, v R, ϑ R x >,.3 then the above Riemann problem admits a contact discontinuity solution, v, ϑx, t = { L, v L, ϑ L x < vt, R, v R, ϑ R x > vt,.4 provided that v L = v R := v, p L = p R := p. The relative entropy method was first introduced by Dafermos [] and Diperna [3] to prove the L stability and uniqueness of Lipschitzian solutions to the conservation laws. The method was used by Leger in [8] to show the stability of shocks in L up to a shift in the scalar case see also [] for an extension to L p, < p <. The method has been extended to the system case in [9] for extreme shocks. The method is purely nonlinear, and allow to work without smallness assumptions. The relative entropy method is also an effective method for the study of asymptotic limits. One of the first use of the method in this context is due to Yau [38] for the hydrodynamic limit of Ginzburg-Landau models. Since then, there have been many works in this context, see [, 3, 4, 5, 5, 3, 3] etc. and the survey paper [36], although they are all concerning the limit to a smooth Lipschitz limit function. Recently, the relative entropy method has been successfully applied in to prove the vanishing viscosity limit of the viscous scalar conservation laws to shocks [], and the L contraction of viscous shock profiles to the scalar viscous conservation laws []. The theory of stability of discontinuous solutions, based on the relative entropy has been reformulated in [33, 3] in terms of contraction, up to a shift. This paper is the first attempt to use this theory for the study of asymptotic limits to a discontinuous solution, in the case of systems. In the present paper, we are concerned with the vanishing physical viscosity limit of compressible Navier-Stokes-Fourier system. to the above contact discontinuity solution.4 to the Euler system., with arbitrarily large wave strength, using the relative entropy method. The contact discontinuities of the Euler system enjoys a contraction property of a special inhomogeneous pseudo-distance, which does not need a shift see [34]. This property is one of the key of our proof.
For the zero dissipation limit of conservation laws, there are many results using BV-theory, see Bianchini-Bressan[6] etc. and the limit to basic wave patterns to the compressible Navier- Stokes equations and Boltzmann equation by the elementary energy method, one can refer to [6, 3, 37, 9,, ] and the references therein. In the present paper, we consider the state equation as in Feireisl [4] p, ϑ = p e + p th, ϑ.5 with p e being the elastic pressure. The thermal pressure p th, ϑ takes the forms as Both functions p e and p ϑ satisfy p th, ϑ = ϑp ϑ. Correspondingly, the internal energy takes the form as where P e is the elastic potential defined by p e, p ϑ, p e, p ϑ..6 e, ϑ = P e + Qϑ,.7 P e = p e ydy, and Qϑ = ϑ C v zdz is the thermal energy satisfying Q ϑ = e ϑ = C v ϑ, where C v ϑ is a positive and smooth function of ϑ. By the second law of thermodynamic we can calculate the entropy s as ϑds = de + pd, ϑ C v z s, ϑ = dz P ϑ,.8 z with P ϑ = p ϑ d. Set U =, v, e, ϑ + v t being the solution to the compressible Navier-Stokes equations v. and Ū =, v, ē, ϑ+ t being the contact discontinuity.4 to the compressible Euler equations.. As usual, the associated relative entropy is defined by su Ū := su sū dsū U Ū, 3
with s being the entropy in.8. Correspondingly, the relative entropy-flux is qu; Ū := dsū fu fū, with the flux fu = v, p, ϑ, p, ϑv t. Define the entropy functional as Et = ϑ su Ūdx..9 Direct calculations show that for the state equations in.5 and.7, one has the explicit formula for.9 [ Et = ϑ su Ūdx = ϑ su + ϑ s Ū + ϑ ds U= Ū U Ū] dx [ ϑ = ϑpϑ ϑ C v z ϑ dz z ϑp ϑ + ϑ C v z dz z + p vv v + E ]dx Ē. [ = ϑ Pϑ P ϑ + p ϑ + P e P e + p e + ϑ Qϑ Q ϑ ϑ C v z dz + ϑ z v v] dx. Now we state the main result as follows. Theorem.. For κ >, let κ, v κ, ϑ κ be a weak solution to. such that su κ Ū lies in L, T ; L R where U κ = κ, v κ, ϑ κ and Ū is the contact discontinuity to the Euler system. and s is defined in.8. Assume further that the pressure p satisfies that or Then if the initial perturbation satisfies that and p e γ, with γ, as +,. p ϑ γ, with γ, as +,. E = O κ, su κ Ū L Ω.3 with Ω R being any neighborhood of x =. Then for any ν, there exists κ > such that if < κ κ, then it holds that Et = O κ, t [, T ] with any fixed T >. Remark.. There is no need of uniform-in-κ L bound on the weak solution U κ, even though its existence is open now. However, if both the initial volume and initial temperature have the lower and upper bound, then existence of global classical solution can be proved following the similar procedure as in Kazhikhov-Shelukhin [4]. 4
Remark.3. By., it can be seen that v κ v = O κ. However, for κ and ϑ κ ϑ, one only can get the convergence in entropy norm as in. and can not derive the convergence in L norm unless one has the uniform lower and upper bound of κ and ϑ κ a priorly. Remark.4. Theorem. shows that the solution of compressible Navier-Stokes system can converge to contact discontinuity solution to the corresponding Euler system with the decay rate κ, uniformly for any viscosity ν, which implies something surprising that the heat conductivity κ dominate the dissipation in the regime of limit of contact discontinuity as κ tends to zero. Note that he wave strength of contact discontinuity can be arbitrarily large and the initial perturbation can contain the possible vacuum states formally. Remark.5. As the state equation, the pressure can contain a large class of gases, such as p, ϑ = Rϑ + p e with R > being the gas constant and the elastic pressure p e satisfying the condition.. Proof of Theorem. In this section, we will prove Theorem. by relative entropy method. We will first define the cut-off entropy functional by suitably choosing the cut-off function ηx based on the underlying structure of contact discontinuity. The cut-off function ηx is chosen to satisfy, x, ηx = smooth and decreasing, x,., x, and For definiteness, we can choose ηx as follows., x, ηx = 4 x3 3 4 x +, x,, x. η x = η x...3 Obviously, we have η x = { 3 4 x 3, 4 x,, otherwise, satisfying η, supp η = [, ] and the symmetry in.. Define the cut-off entropy functional as.4 E t = η x ε ϑ L su U L dx 5 η x ε ϑ R su U R dx.5
where ε > represents the layer-width for the contact discontinuity in the vanishing dissipation limit and is to be determined as ε = κ in the sequel. The functional E t and the cut-off function ηx both play an important role in our analysis. In fact, the following proof is independent of the state equations in.5 and.7 until the estimation.4, that is, all the proofs before.4 are valid to the generic gas satisfying the second law of thermodynamic. By the second law of thermodynamics, it holds that ϑds = de + pd = de v + pd = de vdv + pd, where the total energy E = e, ϑ + v, thus the entropy s satisfies the equation s t = κ ϑ x ϑ x + κ ϑ x ϑ + ν v x ϑ..6 By direct calculations, it holds that de t + = η x dt ε ϑ L + η x ε ϑ R η x ε ϑ L + η x ε ϑ R κ ϑ x ϑ x dx + ν v x ϑ + κ ϑ x ϑ dx η x ε + η x ε pv x v ν v x ϑ x x p x + κ x + ν vv x x pv x dx..7 Note that the third term on the right hand side of.7 vanishes due to the integration by part and the symmetry of the η x. In fact, = η x ε + η x ε pv x v ν v x ϑ x x p x + κ η x ε + η x ε pv v x v ν v x x p p x = η x ε + η x ε +. ε x + ν vv x x pv x dx +κ ϑ x x + ν vv x x pv p v x dx pv v vν v x p p + κϑ x + ν vv x pv p v + η x ε η x ε pv v vν v x p p + κϑ x + ν vv x pv p v dx.8 By integrating.7 with respect to t over [, t] with t [, T ], wow we arrive at E t + = E η x ε ϑ L + η x ε ϑ R t ν v x η x ε ϑ L + η x ε ϑ R 6 ϑ + κ ϑ x ϑ dx κ ϑ x.9 ϑ x dxdt.
Note that the terms on the right hand side of. are non-negative under the condition.6. Then it holds that E t = = η x ε ϑ L su U L dx η x ε [ ϑ L su U L ]dx + [ ϑ L su U L ]dx + + [ ϑ su Ū]dx = Et. η x ε ϑ R su U R dx η x ε [ ϑ R su U R ]dx [ ϑ R su U R ]dx. Substituting the above inequalities.-. into.9, one has [ ϑ Pϑ P ϑ + p ϑ + P e P e + p e + ϑ Qϑ Q ϑ ϑ C v z dz + ϑ z v v] x, tdx + t min{ϑ v L, ϑ R } ν x ϑ + κ ϑ x dxdt ϑ t E η x ε ϑ L + η x ε ϑ R κ ϑ x ϑ x dxdt.. Now we treat the terms on the right hand side of.. First, for the initial values satisfying.3, it holds that E = = + + η x ε ϑ L su U L dx η x ε [ ϑ L su U L ]dx + ε η x ε [ ϑ R su U R ]dx + [ ϑ su Ū]dx + + E + C ε ε η x ε ϑ R su U R dx η x ε [ ϑ L su U L ]dx η x ε [ ϑ R su U R ]dx η x ε dx + C η x ε [ ϑ R su U R ]dx η x ε [ ϑ L su U L ]dx η x dx E + Cε ε. for some generic positive constant C provided that ε = κ is suitably small. Now it remains to control the last term on the right hand side of.. In fact, by integration by part and 7
Cauchy inequality, it holds that t η x ε ϑ L + η x ε ϑ R κ ϑ x ϑ x dxdt t = η x ε ϑ L η x κ ε ϑ ϑ x R ε ϑ dxdt = t ε 4 min{ϑ L, ϑ R } κ ϑ x ϑ dxdt + ϑ L ϑ R min{ϑ L, ϑ R } 4 min{ϑ L, ϑ R } t ε κ ϑ x ϑ dxdt + Cϑ L, ϑ R κ ε t ε κ t ε ε t ε ϑ x η x ε κ ϑ L ϑ R ε ϑ dxdt η x ε dxdt dxdt.3 where the positive constant Cϑ L, ϑ R = 9ϑ L ϑ R 6 min{ϑ L,ϑ R }. With the notation { } being the characteristic function of the set { } R [, T ] and C being suitably large positive constant to be determined, it holds that t ε C ε + dxdt = t ε t ε { C } + { >C } dxdt.4 { >C } dxdt. Now the state equations in.5 and.7 as in [4] are crucially used to control the second term on the right hand of.4. Moreover, if the pressure function p e satisfies. or p ϑ satisfies., then P e or P ϑ γ, γ, as +. Therefore, we can choose the suitably large positive constant C such that ϑ P ϑ P ϑ + p ϑ + P e P e + p e { >C < +,.5 } therefore, one has t ε C { >C } dxdt [ ϑ Pϑ P ϑ + p ϑ t + P e P e + p e ] dxdt. Substituting.6 into.4 and then into.3 yields that t η x ε ϑ L + η x ε ϑ R κ ϑ x ϑ x dxdt t ε 4 min{ϑ L, ϑ R } κ ϑ x ϑ dxdt + Cϑ L, ϑ R κ t [ ϑ Pϑ ε P ϑ +p ϑ + P e P e + p e ] dxdt..6.7 8
The combination of.,. and.7 implies that Set Et + 4 min{ϑ L, ϑ R } t v ν x ϑ + κ ϑ x dxdt ϑ E + Cε + C κ t ε ε + Etdt. F t = t Etdt, then applying Gronwall inequality to.8 implies that F t C ε + E + ε for some positive constant C. Furthermore, one has κ ε Et C [E + ε + κ ε ε +.8 e C κ ε t,.9 ε + E + ε κ ε e C κ ε t].. Choosing the contact discontinuity layer ε = κ and the initial perturbation E = O κ as in Theorem., one has which completed the proof of Theorem.. References Et = O κ, t [, T ], ν, [] Adimurthi, Sh. S. Goshal, G. D Veerappa Gowda, L p stability for entropy solutions of scalar conservation laws with strict convex flux, J. Differential Equations, 56 4, no., 3395 346. [] C. Bardos, F. Golse, D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statist. Phys. 63 99, no. -, 33-344. [3] C. Bardos, F. Golse, D. Levermore, Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46 993, no. 5, 667-753. [4] F. Berthelin, A. Tzavaras, A. Vasseur, From discrete velocity Boltzmann equations to gas dynamics before shocks. J. Stat. Phys. 35 9, no., 53-73. [5] F. Berthelin, A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36 5, no. 6, 87-835. [6] S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math., 6 5, 3-34. 9
[7] G. Q. Chen, H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 47 999, 89-9. [8] G. Q. Chen, H. Frid, Large-time behavior of entropy solutions of conservation laws, J. Differential Equations, 5 999, 38-357. [9] G. Q. Chen, H. Frid, Y. C. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Comm. Math. Phys., 8, -7. [] K. Choi, A. Vasseur, Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method, SIAM J. Math. Anal., 47 5, 45-48. [] C. M. Dafermos, The second law of thermodynamics and stability, Arch. Ration. Mech. Anal.,7 979, 67-79. [] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. 35, Springer-Verlag, Berlin,. [3] R. J. Diperna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., 8 979, 38-88. [4] E. Feireisl, Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 6. Oxford University Press, Oxford, 4. [5] F. Golse, L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 55 4, no., 8-6. [6] J. Goodman, Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 99, 35-65. [7] H. David, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states. Z. Angew. Math. Phys., 49 998, no. 5, 774-785. [8] D. Hoff, T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38 989, 86-95. [9] F. M. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity, Kinetic and Related Models, 3, 685-78. [] F. M. Huang, Y. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier- Stokes equations for solutions to Riemann problem, Arch. Ration. Mech. Anal., 3, 379-43. [] Feimin Huang, Yi Wang, Yong Wang, Tong Yang, The limit of the Boltzmann equation to the Euler equations, SIAM J. Math. Anal., 45 3, no.3, 74-8. [] M. J. Kang, A. Vasseur, L contraction for shock waves of scalar viscous conservation laws, Preprint.
[3] M. J. Kang, A. Vasseur, Criteria on contractions for entropic discontinuities of systems of conservation laws. Preprint. [4] A. V. Kazhikhov, V. V. Shelukhin, Unique global solution with respect to time of initialboundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech., 4 977, no., 73-8.; translated from Prikl. Mat. Meh., 4 977, no., 8-9. [5] Young-Sam Kwon, Asymptotic limit to shocks for scalar balance laws using relative entropy. Abstr. Appl. Anal. 4, Art. ID 698, 7. [6] Y.-S. Kwon, A. Vasseur, Asymptotic limit to a shock for BGK models using the relative entropy method, Nonlinearity, 8 5 53-543. [7] S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids, SIAM J. Math. Anal., 38 6, 368-384. [8] N. Leger, L stability estimates for shock solutions of scalar conservation laws using the relative entropy method. Arch. Ration. Mech. Anal. 99, no. 3, 76-778. [9] N. Leger, A. Vasseur, Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-bv perturbations. Arch. Ration. Mech. Anal., no., 7-3 [3] P. L. Lions, N. Masmoudi, From the Boltzmann equations to the equations of incompressible fluid mechanics. I, II. Arch. Ration. Mech. Anal. 58, no. 3, 73-93, 95-. [3] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the -D compressible Navier-Stokes equations, J. Differential Equations, 48, 95-. [3] N. Masmoudi, L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math. 56 3, no. 9, 63-93. [33] D. Serre, A. Vasseur, L -type contraction for systems of conservation laws, Journal de l École polytechnique Mathématiques, 4, -8. [34] D. Serre, A. Vasseur, About the relative entropy method for hyperbolic systems of conservation laws, Preprint 5. [35] J. Smoller, Shock Waves and Reaction-Diffusion Equations, nd ed., New York: Springer- Verlag, xxii, 994. [36] A. Vasseur, Recent results on hydrodynamic limits. Handbook of differential equations: evolutionary equations. Vol. IV, 33376, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 8. [37] Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock, Acta Mathematica Scientia, 8B 8, 77-748.
[38] H.-T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 99, 63-8.